Archimedean field

This article defines a field property: a property that can be evaluated to true/false for any field.
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Definition

In terms of a subfield of the reals

An Archimedean field is a field that is isomorphic to a subfield of the field of real numbers.

In terms of a total ordering

An Archimedean field is a field with a total ordering $\le$ on its elements satisfying the following:

• $0\le 1$.
• $a\le b⟹-b\le -a$.
• $a\le b$ and $c\le d$ implies that $a+c\le b+d$.
• $0\le a$ and $b\le c$ implies that $ab\le ac$.
• The Archimedean property: For any $x\in K$, there exists a natural number $n$ such that $x\le n$, where the natural number $n$ is viewed as the element of $K$ obtained by adding $1$ to itself $n$ times.

Relation with other properties

Weaker properties

• Formally real field: For proof of the implication, refer Archimedean implies formally real and for proof of its strictness (i.e. the reverse implication being false) refer Formally real not implies Archimedean.
• Field of characteristic zero

Incomparable properties

• Euclidean field
• Pythagorean field